The present application is directed to dielectrostrictive sensors and their use in measuring the characteristics of viscoelastic materials
Dielectrostriction is a rheo-dielectric phenomenon reflecting the change of dielectric properties of a material under deformation or during flow, and is a fundamental property of any dielectric. It has been found that a linear relationship exists between the change of dielectric constant and stress, which is called the stress-dielectric rule. Also the strain-dielectric and stress-dielectric coefficients have been found sensitive to material structure and composition, particle dispersion, and particle size distribution, degree of elasticity, degradation, temperature, and moisture, etc.
Solid state sensors have been developed that detect deformation based on the dielectrostrictive response of the sample under test. See, e.g. U.S. Pat. No. 7,726,199, which is incorporated herein by reference. As described and illustrated therein, a dielectrostrictive sensor has a planar configuration with interdigitated electrodes deposited on a non-conductive substrate. During a use, the material being tested is adjacent to electrodes, but does not necessarily touch them. The deformation of the test material changes the dielectric properties of the material, which, therefore, changes the sensor capacitance.
Once the stress-dielectric coefficient of the test material is determined, the stress can be obtained through the change of dielectric constant, and therefore, through the change of the sensor's capacitance based on the stress-dielectric rule. On the other hand, once the strain or stress is known, the strain-dielectric coefficients, or the stress-dielectric coefficients, can be obtained, which can be used for process monitoring and material quality examination.
Based on the stress-dielectric rule, a linear relationship exists between the change of a dielectric constant and stress a and can be expressedas,εij=εδij+λ1σij+λ2σllδij,  (1)where ε is the dielectric constant of the material before deformation, and λ1 and λ2 are stress-dielectric coefficients, and δij the Kronecker delta.
For a viscoelastic material when the change of dielectric constant is small, the change of the dielectric constant can also be expressed in terms of strain u, as,εij=εδij+α1uij+α2ullδij,  (2)and α1 and α2 are strain-dielectric coefficients.
With reference to FIGS. 1A, 1B and 1C, a dielectrostrictive sensor is formed by inter-digitated electrodes having equal width, W=2w, and being separated by a distance, A=2a, which is attached to an originally isotropic material with dielectric constant ε. Both the thickness of the dielectric material, h (not shown), and the length of the electrode, l, are much larger than the electrode width and the separation (h,l>>W, A). The electrodes in FIG. 1 are located in the xy-plane and form an angle, θ, with respect to the y axis. The capacitance, Cij of such a planar sensor is,
                                                        C              θ                        =                                          C                0                            ⁡                              (                                                      ɛ                    effective                                    +                                      ɛ                    s                                                  )                                              ,                                          ⁢                                    C              0                        =                                                                                ɛ                    0                                    ⁢                  L                                π                            ⁢                              ln                ⁡                                  (                                      1                    +                                          w                      a                                                        )                                                              ,                                          ⁢          and                ⁢                                  ⁢                                            ɛ              effective                        =                                          [                                                                                                                                                          (                                                                                                                            ɛ                                  xx                                                                ⁢                                                                  ɛ                                  zz                                                                                            -                                                              ɛ                                xz                                2                                                                                      )                                                    ⁢                                                      cos                            2                                                    ⁢                          θ                                                +                                                                              (                                                                                                                            ɛ                                  yy                                                                ⁢                                                                  ɛ                                  zz                                                                                            -                                                              ɛ                                yz                                2                                                                                      )                                                    ⁢                                                      sin                            2                                                    ⁢                          θ                                                +                                                                                                                                                2                        ⁢                                                  (                                                                                                                    ɛ                                xy                                                            ⁢                                                              ɛ                                zz                                                                                      -                                                                                          ɛ                                xz                                                            ⁢                                                              ɛ                                yz                                                                                                              )                                                ⁢                        cos                        ⁢                                                                                                  ⁢                        θ                        ⁢                                                                                                  ⁢                        sin                        ⁢                                                                                                  ⁢                        θ                                                                                            ]                                            1                /                2                                              ,                                    (        3        )            where εs is the dielectric constant of the sensor substrate and εeffective is the effective dielectric constant of the material on the top. For an isotropic material, εeffective=ε, therefore, the capacitance of the sensor before deformation is Cθ=C0 (ε+εs). L is the total length of all electrode strips, and 2C0 represents the capacitance of the electrodes in the free space. When a=w, C0 is estimated to be
                    ɛ        0            ⁢      L        π    ⁢      ln    ⁡          (              1        +                  w          a                    )      from Eq. (3). The value C0 also can be directly measured for any given electrode pattern.
With reference to FIG. 1A, a dielectric material adjacent to electrodes is under a shear deformation or experiences a shear flow, uyz=yz, inter-digitated electrodes are located in the xy-plane and form an angle, θ, with the y-axis. In FIG. 1B, the dielectric material is subjected to shear stress σyz and might also be subjected to normal stresses σyz, σxx, σyy and σzz, if the material is viscoelastic.
When the dielectric material is under a deformation, or experiences a flow, it becomes anisotropic and its dielectric constant becomes a second rank tensor. As an example, if the dielectric material is under a shear deformation or experiences shear flow, Uy=yz, as shown in FIG. 1A, the non-zero strain components to which it is subjected would be
                    u        yz            =              γ        2              ,                  ⁢    and                      u        zz            =                        γ          2                2              ,  and the non-zero stress components would be shear stress σyz, and there might also be normal stresses, σxx, σyy, and σzz, if the material is a viscoelastic, as shown in FIG. 1B.
According to the stress-dielectric relationship in Eq. (1), the dielectric constant tensor can be expressed in terms of stress,
                              ɛ          =                      [                                                                                                                                                        ɛ                          +                                                                                    λ                              1                                                        ⁢                                                          σ                              xx                                                                                +                                                                                                                                                                                          λ                            2                                                    ⁢                                                      σ                            ll                                                                                                                                                                0                                                  0                                                                              0                                                                      ɛ                    +                                                                  λ                        1                                            ⁢                                              σ                        yy                                                              +                                                                  λ                        2                                            ⁢                                              σ                        ll                                                                                                                                                        λ                      1                                        ⁢                                          σ                      yz                                                                                                                    0                                                                                            λ                      1                                        ⁢                                          σ                      yz                                                                                                                                                                                    ɛ                          +                                                                                    λ                              1                                                        ⁢                                                          σ                              zz                                                                                +                                                                                                                                                                                          λ                            2                                                    ⁢                                                      σ                            ll                                                                                                                                                                    ]                          ,                            (        4        )            where σll=σxx+σyy+σzz. Substituting the components of the dielectric constant tensor into Eq. (3) yields the expression for the capacitance of the planar dielectrostrictive sensor in terms of the stress after deformation,
                              C          θ                =                                            C              0                        ⁡                          (                                                                                          ɛ                      +                                                                        1                          2                                                ⁢                                                  (                                                                                                                                                                                                                                λ                                      1                                                                        ⁢                                                                          σ                                      xx                                                                        ⁢                                                                          cos                                      2                                                                        ⁢                                    θ                                                                    +                                                                                                                                                                                                                                                                                                λ                                      1                                                                        ⁢                                                                          σ                                      yy                                                                        ⁢                                                                          sin                                      2                                                                        ⁢                                    θ                                                                    +                                                                                                                                                                                                                                                                                                λ                                      1                                                                        ⁢                                                                          σ                                      zz                                                                                                        +                                                                      2                                    ⁢                                                                                                                                                  ⁢                                                                          λ                                      2                                                                        ⁢                                                                          σ                                      ll                                                                                                                                                                                                                    )                                                                    +                                                                                                                                                                                                                                              (                                                                                                                                                                                                                                                            λ                                          1                                                                                ⁢                                                                                  σ                                          xx                                                                                ⁢                                                                                  cos                                          2                                                                                ⁢                                        θ                                                                            +                                                                                                                                                                                                                                                                                                                                    λ                                          1                                                                                ⁢                                                                                  σ                                          yy                                                                                ⁢                                                                                  sin                                          2                                                                                ⁢                                        θ                                                                            +                                                                                                                                                                                                                                                                                          λ                                        2                                                                            ⁢                                                                              σ                                        ll                                                                                                                                                                                                        )                                                        ⁢                                                          (                                                                                                                                    λ                                    1                                                                    ⁢                                                                      σ                                    zz                                                                                                  +                                                                                                      λ                                    2                                                                    ⁢                                                                      σ                                    ll                                                                                                                              )                                                                                -                                                                                    λ                              1                              2                                                        ⁢                                                          σ                              yz                              2                                                        ⁢                                                          sin                              2                                                        ⁢                            θ                                                                                                    2                          ⁢                                                                                                          ⁢                          ɛ                                                                    +                                                                                                                                  ɛ                      s                                                                                  )                                .                                    (        5        )            
According to the strain-dielectric relationship in Eq. (2), the dielectric constant tensor can be expressed in terms of strain,
                    ɛ        =                  [                                                                      ɛ                  +                                                                                    γ                        2                                            2                                        ⁢                                          α                      2                                                                                                  0                                            0                                                                    0                                                              ɛ                  +                                                                                    γ                        2                                            2                                        ⁢                                          α                      2                                                                                                                                        γ                    2                                    ⁢                                      α                    1                                                                                                      0                                                                                  γ                    2                                    ⁢                                      α                    1                                                                                                ɛ                  +                                                                                    γ                        2                                            2                                        ⁢                                          (                                                                        α                          1                                                +                                                  α                          2                                                                    )                                                                                                    ]                                    (        6        )            
Substituting the components of the above dielectric constant tensor into Eq. (3) yields the capacitance of the planar dielectrostrictive sensor in terms of the strain after deformation,
                              C          θ                =                                            C              0                        2                    ⁢                                    (                              ɛ                +                                                                                                    2                        ⁢                                                                                                  ⁢                        ɛ                        ⁢                                                                                                  ⁢                                                  α                          1                                                                    +                                              4                        ⁢                                                                                                  ⁢                        ɛ                        ⁢                                                                                                  ⁢                                                  α                          2                                                                    -                                                                        α                          1                          2                                                ⁢                                                  sin                          2                                                ⁢                        θ                                                                                    8                      ⁢                                                                                          ⁢                      ɛ                                                        ⁢                                      γ                    2                                                  +                                  ɛ                  s                                            )                        .                                              (        7        )            
According to Eqs. (5) and (7), by extracting signals from two differently oriented sensors (θ1 and θ2 with respect to the y-direction), one can obtain the capacitance difference in terms of strain,
                                                        C                              θ                ⁢                                                                  ⁢                1                                      -                          C                              θ                ⁢                                                                  ⁢                2                                              =                                                    C                0                                            16                ⁢                                                                  ⁢                ɛ                                      ⁢                          α              1              2                        ⁢                                          γ                2                            ⁡                              (                                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      1                                                        -                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      2                                                                      )                                                    ,                            (        8        )            or in terms of stress,
                                          C                          θ              ⁢                                                          ⁢              1                                -                      C                          θ              ⁢                                                          ⁢              2                                      =                                                            C                0                                            4                ⁢                                                                  ⁢                ɛ                                      ⁡                          [                                                                    (                                                                  σ                        xx                                            -                                              σ                        yy                                                              )                                    ⁢                                      (                                          ɛ                      +                                                                        σ                          zz                                                ⁢                                                  λ                          1                                                                    +                                                                        σ                          ll                                                ⁢                                                  λ                          2                                                                                      )                                                  +                                                      σ                    yz                    2                                    ⁢                                      λ                    1                                                              ]                                ⁢                                                    λ                1                            ⁡                              (                                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      1                                                        -                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      2                                                                      )                                      .                                              (        9        )            
If the normal stress contribution, σxx−σyy, in Eq, (9), does not exist, or is much less than the shear stress contribution, σyz2λ1, it can therefore be neglected, then Eq, (9) can reduce to
                                          C                          θ              ⁢                                                          ⁢              1                                -                      C                          θ              ⁢                                                          ⁢              2                                      =                                            C              0                                      4              ⁢              ɛ                                ⁢                      σ            yz            2                    ⁢                                                    λ                1                2                            ⁡                              (                                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      1                                                        -                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      2                                                                      )                                      .                                              (        10        )            
For Material showing a viscoplastic behavior, deformed or flow sheared above yield stress σY, the capacitance difference between the two differently oriented sensors are,
                                          C                          θ              ⁢                                                          ⁢              1                                -                      C                          θ              ⁢                                                          ⁢              2                                      =                                            C              0                                      4              ⁢              ɛ                                ⁢                                    (                                                σ                  yz                                -                                  σ                  Y                                            )                        2                    ⁢                                                    λ                1                2                            ⁡                              (                                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      1                                                        -                                      cos                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                                          θ                      2                                                                      )                                      .                                              (        11        )            
To measure the strain-dielectric coefficient, α1, and the stress-dielectric coefficient, λ1, it is known to use a planar sensor rosette with at least two differently orientated sensors that are oriented at different directions, θ1, and θ2 with the y-axis, as shown in FIG. 2.
When the original material dielectric constant, ε, orientation angles, θ1, and θ2, engineering strain, γ, shear stress, σyz, are known, the strain-dielectric coefficient, α1, and the stress-dielectric coefficient, λ1, can be determined by obtaining the capacitance difference, Cθ1-Cθ2.
If the normal stress contribution cannot be neglected, detection of the first stress-dielectric coefficient, λ1, requires the information of the first and second normal stress differences, −(N1+N2)=σxx−σyy, and dielectric constant component, εzz=ε+σzzλ1+σllλ2. The values of the first and second normal stress differences can be measured with a rheometer with cone-plate and parallel-plate configurations. The dielectric constant component, εzz, could be obtained through measuring the capacitance of two metal sheets attached to the side surfaces of the inner and outer cylinders of the rheometer with a co-axial cylindrical system during the shear flow. When the gap between the inner and outer cylinders is much smaller than their radii, approximation can be assumed, and the two metal sheets with the testing material in between can be treated as a parallel-plate capacitor.
By way of the present application, a sensor rosette is provided that allows the measurement of shear stress in a viscoelastic material and methods are provided for measuring the compositions and properties or detecting changes in compositions and properties of viscoelastic substances having known dielectric constants before deformation.